Normalized defining polynomial
\( x^{10} - 4x^{9} - 9x^{8} + 45x^{7} + 3x^{6} - 124x^{5} + 56x^{4} + 77x^{3} - 27x^{2} - 15x + 2 \)
Invariants
Degree: | $10$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
| |
Signature: | $[10, 0]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
| |
Discriminant: | \(1630033565404672\) \(\medspace = 2^{9}\cdot 47^{3}\cdot 313^{3}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(33.21\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | $2^{3/2}47^{1/2}313^{1/2}\approx 343.0568466012594$ | ||
Ramified primes: | \(2\), \(47\), \(313\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q(\sqrt{29422}) \) | ||
$\card{ \Aut(K/\Q) }$: | $1$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{278}a^{9}-\frac{6}{139}a^{8}-\frac{26}{139}a^{7}+\frac{22}{139}a^{6}-\frac{71}{278}a^{5}-\frac{56}{139}a^{4}-\frac{21}{278}a^{3}+\frac{53}{139}a^{2}+\frac{49}{139}a-\frac{52}{139}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
Unit group
Rank: | $9$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
| |
Fundamental units: | $\frac{97}{139}a^{9}-\frac{469}{139}a^{8}-\frac{1053}{278}a^{7}+\frac{4963}{139}a^{6}-\frac{3412}{139}a^{5}-\frac{10864}{139}a^{4}+\frac{28035}{278}a^{3}-\frac{282}{139}a^{2}-\frac{8093}{278}a+\frac{1032}{139}$, $\frac{233}{278}a^{9}-\frac{989}{278}a^{8}-\frac{915}{139}a^{7}+\frac{5404}{139}a^{6}-\frac{2087}{278}a^{5}-\frac{27069}{278}a^{4}+\frac{19015}{278}a^{3}+\frac{10381}{278}a^{2}-\frac{3178}{139}a-\frac{579}{139}$, $\frac{257}{278}a^{9}-\frac{1277}{278}a^{8}-\frac{1271}{278}a^{7}+\frac{6766}{139}a^{6}-\frac{10463}{278}a^{5}-\frac{29757}{278}a^{4}+\frac{20584}{139}a^{3}-\frac{141}{278}a^{2}-\frac{13317}{278}a+\frac{814}{139}$, $\frac{120}{139}a^{9}-\frac{606}{139}a^{8}-\frac{1221}{278}a^{7}+\frac{6531}{139}a^{6}-\frac{4628}{139}a^{5}-\frac{15108}{139}a^{4}+\frac{36799}{278}a^{3}+\frac{1739}{139}a^{2}-\frac{11647}{278}a+\frac{586}{139}$, $\frac{243}{139}a^{9}-\frac{2079}{278}a^{8}-\frac{1794}{139}a^{7}+\frac{11109}{139}a^{6}-\frac{3492}{139}a^{5}-\frac{51513}{278}a^{4}+\frac{23114}{139}a^{3}+\frac{8565}{278}a^{2}-\frac{7044}{139}a+\frac{721}{139}$, $\frac{65}{139}a^{9}-\frac{224}{139}a^{8}-\frac{1339}{278}a^{7}+\frac{2443}{139}a^{6}+\frac{1084}{139}a^{5}-\frac{6307}{139}a^{4}+\frac{2691}{278}a^{3}+\frac{3137}{139}a^{2}-\frac{465}{278}a-\frac{366}{139}$, $\frac{162}{139}a^{9}-\frac{693}{139}a^{8}-\frac{1196}{139}a^{7}+\frac{7406}{139}a^{6}-\frac{2328}{139}a^{5}-\frac{17171}{139}a^{4}+\frac{15363}{139}a^{3}+\frac{2855}{139}a^{2}-\frac{4418}{139}a+\frac{249}{139}$, $\frac{57}{278}a^{9}-\frac{267}{278}a^{8}-\frac{323}{278}a^{7}+\frac{1393}{139}a^{6}-\frac{1823}{278}a^{5}-\frac{5967}{278}a^{4}+\frac{3780}{139}a^{3}+\frac{65}{278}a^{2}-\frac{2337}{278}a+\frac{94}{139}$, $\frac{75}{139}a^{9}-\frac{344}{139}a^{8}-\frac{989}{278}a^{7}+\frac{3717}{139}a^{6}-\frac{1711}{139}a^{5}-\frac{8817}{139}a^{4}+\frac{17561}{278}a^{3}+\frac{1834}{139}a^{2}-\frac{6011}{278}a+\frac{262}{139}$ | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
| |
Regulator: | \( 41001.9964635 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{10}\cdot(2\pi)^{0}\cdot 41001.9964635 \cdot 1}{2\cdot\sqrt{1630033565404672}}\cr\approx \mathstrut & 0.519968088727 \end{aligned}\]
Galois group
A non-solvable group of order 120 |
The 7 conjugacy class representatives for $S_5$ |
Character table for $S_5$ |
Intermediate fields
The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
Degree 5 sibling: | 5.5.117688.1 |
Degree 6 sibling: | 6.6.1630033565404672.1 |
Degree 10 sibling: | data not computed |
Degree 12 sibling: | data not computed |
Degree 15 sibling: | data not computed |
Degree 20 siblings: | data not computed |
Degree 24 sibling: | data not computed |
Degree 30 siblings: | data not computed |
Degree 40 sibling: | data not computed |
Minimal sibling: | 5.5.117688.1 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | ${\href{/padicField/7.5.0.1}{5} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.5.0.1}{5} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.3.0.1}{3} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | R | ${\href{/padicField/53.5.0.1}{5} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | $\Q_{2}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
2.3.0.1 | $x^{3} + x + 1$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
2.6.9.5 | $x^{6} + 12 x^{5} + 68 x^{4} + 226 x^{3} + 457 x^{2} + 514 x + 243$ | $2$ | $3$ | $9$ | $C_6$ | $[3]^{3}$ | |
\(47\) | $\Q_{47}$ | $x + 42$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
47.3.0.1 | $x^{3} + 3 x + 42$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
47.6.3.2 | $x^{6} + 147 x^{4} + 84 x^{3} + 6636 x^{2} - 11592 x + 92756$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
\(313\) | $\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
$\Q_{313}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ |